5 pages matching constraint equations in this book
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associated auxiliary dual function auxiliary dual problem auxiliary primal problem auxiliary variable canonical form chemical elements chemical equilibrium problem chemical potential chemical system components concave function constraints Eq convex convex function degrees of difficulty domain defined dual constraints dual variables duality relation Duffin-Zener type equal equations equilibrium constants equivalent exponents free energy Geometric Inequality geometric programming problem given by Eq Hessian matrix homogeneous function independent variable Lagrange multipliers Lagrangian form Lemma mathematical programming maximization maximum member of Eq minimization problem minimum necessary conditions non-negative notation number of primal optimal optimum original problem orthogonality polynomials possible primal canonical primal constraints primal function primal objective function primal program primal variables probability of satisfactory product function Proof pseudo pseudo-maximum point pseudo-optimum points reactor saddle point satisfactory operation satisfies the necessary satisfy Eq signum function solution solved species stationary points transformed vector zero degrees